This will be boring, useless, and wrong, most likely – but here’s a walkthrough of what I got and what I did with it so far. Click on any photo to see a larger version, but be warned that they will all be fuzzy.
This is my battery:

It’s an Elecsol 110 amp 12 volt. I am probably not capitalising things I should be. It is slightly too heavy for me, and I should have got a 100 amp. Ultimately I want gel batteries (I think – but possibly I mean glass mat batteries…), but that’s on the five year plan. On the one-year plan is one more battery just like this, or possibly two. You’re allowed to mix new and old batteries, but not different size batteries, as I currently understand things, so I can add more whenever, as long as it’s another 110 amp.
The little nubbins/terminals came in a naked state, and I needed to then order the terminal clamps which can be seen here, which allow things to be attached to the battery. One bit of the clamp goes on the nubbin and gets its bolt tightened so it doesn’t come off, and at the other bit of the clamp is a threaded bolt with a sort of moebius strip of a washer and a nut.
Apparently, as long as I don’t touch both terminals at once, or allow anything conductive to touch both terminals at once, I’m not going to cause any problems, death-wise. Someone on a canal forum made a comment about someone doing stupid things with electricity, saying something like ‘we’ll see if you’re laughing when you (blah get zapped or something) – DC has a way of making you stick to it, not like AC which throws you clear’. Which put the fear of God into me, somewhat. Not that I was lacking this in the first place. Oddly, the only warnings I see about all this seem to be ‘don’t allow metal things to come into contact with both terminals simultaneously, as this will damage the battery’, and that’s it.
This is my regulator and the wires that go from the battery to it:

Hm. It’s upside-down and it won’t rotate even though I’ve told it to, but whatever.
The regulator is necessary for a few reasons, two of which are – it protects the battery from overcharging/overdischarging (?), and it has a diode in it which stops electricity running back out of the battery when it’s not being charged. Also useful is that it has connection places for the bit that I can plug things into. Which is called an inverter for now, since I want to be able to plug in normal 240 volt AC appliances, not just 12 volt DC, but at some point I’ll want the 12 volt stuff as well, and will need to figure this out.
If you blow up the picture of the regulator, you’ll see (fuzzily) that there are sort of rings at one end of those wires. The ring bits go over the bolts on the battery terminal – they’re slightly too small, and needed to be forced to screw on and around the threads, which deformed them a bit, but I’m going to file that under ‘feature’. I threaded the red/positive wire onto the battery first, then placed the washer and tightened down the bolt. Then I fiddled with the other end, which had about 1 cm of exposed small twisted copper wires, and I sort of split them in the middle so they could fit half and half into the area in the reguator where they were meant to go – between two plates with a screw in the middle that tightens down to hold the wire in place. Then I did the same thing with the black/negative wire – screwed the loop end onto the battery terminal (after fiddling with the regulator end of the wire to get it ready to slide in easily) – then put the regulator end into the regulator and screwed that down. (The regulator had instructions about which bits to connect in what order, and into which slots, and for backup/instruction-reading-impaired folks, there are images on the front of it as well.) At this point the little lights on the regulator lit up to tell me the battery was 2 lights out of three charged, and that there was a load on the battery (presumably the regulator lights).

This is the back of my solar panel, with its wires:

So next I connected the red/positive wire from the solar panel to the regulator, and finally the same with the black/negative one. As soon as I moved the panel to the open back door, a light on the regulator came on telling me the battery was charging.

It was a grey day, raining some, and the panel was at that point totally vertical, still indoors, and facing a fence, wall, and trees. So, this is heartening, that it will charge (even a tiny bit) on so little input.
This is the solar panel all hooked up and being useful:

Next to do is to sort the inverter I got, which came with a cigarette lighter adapter on the end instead of bare wires – I had seen that it came with this, but thought it was an option that was included for if you wanted it, not like, what it was intended for, since there was another product on the website that looked like it was a dedicated car-adapter thing. So that should be easy enough to sort, and then I can connect that to the regulator in the same way, and then I’ll be able to plug in a normal applicance such as a computer or phone or possibly kettle, but I’ll have to look that up – anything under 150 watts, which rules out the microwave.
In the one-year-or-under plan is another battery (or two), another panel (maybe same kind, maybe a low-profile peel-and-stick), a different regulator which should get more power from the panel to the battery/ies, a meter of some kind so I can see actual amounts of what’s coming in from the panel and what’s coming out of the battery, and a more powerful inverter, possibly – but I’ll have to think more about that later.
Other notes: the batteries need to be kept someplace well-ventilated, and ideally in some sort of tray/container thing so if they leak, the acid is contained – I’ll need to look up more information on this and see what’s good for neutralising and how one deals with leaks/spills. This is why I want the gel kind next – no leaks (if I’m getting the type correct).
eta – hee. Big NO on the kettle – it’s much much worse than the microwave. Which is ok – the whole focus just now was computers and phones, so I can even stick to DC for most of that, and there’s a cheap thingy I can get that will dial up the specific correct voltage for each of my little addictions,which might in turn actually solve one of my computer problems, if I’m extra lucky! And no inverter needed for that, which is good cos they eat up some of the power in converting the current (or whatever is the correct way of saying it).